Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-19T19:35:22.937Z Has data issue: false hasContentIssue false
  • English
  • Français

Stepping-Stone Model with Circular Brownian Migration

Published online by Cambridge University Press:  20 November 2018

Xiaowen Zhou
Xiaowen Zhou*
Affiliation:
Department of Mathematica and Statistics, Concordia University, Montreal, QC, H3G 1M8 e-mail: xzhou@mathstat.concordia.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider the stepping-stone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow on the circle and the marginal distribution of this model. We then give a new representation for the stepping-stone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computations. In particular, we find the distribution for the first time when there is only one type left across the circle.

Keywords

60G5760J65stepping-stonemodelcircular coalescing Brownian motionArratia flowdualityentrance Law
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 146 - 160
Copyright
Copyright © Canadian Mathematical Society 2008

References

[Arr79] Arratia, R., Coalescing Brownian motions on the line. Ph.D. thesis, University of Wisconsin, Madison, 1979.Google Scholar
[DEFKZ00] Donnelly, P., Evans, S. N., Fleischmann, K., Kurtz, T. G., and Zhou, X., Continuum-sites stepping-stone models, coalescing exchangeable partitions, and random trees. Ann. Probab. 28(2000), no. 3, 1063–1110.Google Scholar
[DK96] Donnelly, P. and Kurtz, T. G., A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24(1996), no. 2, 698–742.Google Scholar
[Eva97] Evans, S. N., Coalescing markov labeled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. H. Poincaré Probab. Statist. 33(1997), no. 3, 339–358.Google Scholar
[Har84] Harris, T. E., Coalescing and noncoalescing stochastic flows in R 1 . Stochastic Process. Appl. 17(1984), no. 2, 187–210.Google Scholar
[JR05] Le Jan, Y. and Raimond, O., Flows, coalescence and noise. Ann. Probab. 32(2004), no. 2, 1247–1315.Google Scholar
[Kal76] Kallenberg, O., Random Measures. Academic Press, London, 1976.Google Scholar
[Kim53] Kimura, M., “Stepping-Stone” Models of Population. Technical Report 3, Institute of Genetics, Japan, 1953.Google Scholar
[Kni81] Knight, F. B., Essentials of BrownianMotion and Diffusion. Mathematical Surveys 18, American Mathematical Society, Providence, RI, 1981.Google Scholar
[Lig85] Liggett, T. M., Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 227, Springer-Verlag, New York, 1985.Google Scholar
[Mat89] Matsumoto, H., Coalescing stochastic flows on the real line. Osaka J. Math. 26(1989), no. 1, 139–158.Google Scholar
[RY91] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293, Springer-Verlag, Berlin, 1991.Google Scholar
[Shi88] Shiga, T., Stepping stone models in population genetics and population dynamics. In: Stochastic Processes in Physics and Engineering. Math. Appl. 42, Riedel, Dordrecht, 1988, 345–355.Google Scholar
[Zho07] Zhou, X., A superprocess involving both branching and coalescing. Ann. Inst. H. Poincaré Probab. 43(2007) 599–618.Google Scholar
[Zho03] Zhou, X., Clustering behavior of a continuous-sites stepping-stone model with Brownian migration. Electron. J. Probab. 8(2003), no. 11 (electronic).Google Scholar